SOLUTION: Consider the function f(x)=3x-5 and let f^n=fofofof....of where the composition is done n times. Find h(x)=f^n(x)

Algebra ->  Functions -> SOLUTION: Consider the function f(x)=3x-5 and let f^n=fofofof....of where the composition is done n times. Find h(x)=f^n(x)      Log On


   

Question 354514: Consider the function f(x)=3x-5 and let f^n=fofofof....of where the composition is done n times. Find h(x)=f^n(x)
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
f%5En%28x%29 = %283%5En%29x-5(1%2B3%2B3%5E2+...+3%5E%28n-1%29).
This can be proved by induction:
If n=1, then f%5E1%28x%29=f%28x%29=3x-5%2A3%5E0.
If n=2, then f%5E2%28x%29=f%28x%29=3%283x-5%29x-5+=+%283%5E2%29x-3%2A5-5=%283%5E2%29x-5%2A%281%2B3%5E1%29.
Assume for the purpose of induction that it is true for n+=+k. have to show that it is also true for n=k.
Assume this is true:
f%5E%28k-1%29%28x%29 = %283%5E%28k-1%29%29x-5(1%2B3%2B3%5E2+...+3%5E%28k-2%29).
Then f%5Ek%28x%29+=+f%5E%28k-1%29%28f%28x%29%29,and
f%5Ek%28x%29=+f%5E%28k-1%29%283x-5%29,
f%5Ek%28x%29+=3%5E%28k-1%29%283x-5%29-5(1%2B3%2B3%5E2+...+3%5E%28k-2%29).
f%5Ek%28x%29+=%283%5Ek%29x-5%2A3%5E%28k-1%29-5(1%2B3%2B3%5E2+...+3%5E%28k-2%29) by the distributive property.
f%5Ek%28x%29+=%283%5Ek%29x-5(1%2B3%2B3%5E2+...+3%5E%28k-2%29%2B3%5E%28k-1%29) by associativity.
Therefore by induction the formula for f%5En+%28x%29 is true for all positive integral n by induction. Now
f%5En%28x%29 = %283%5En%29x-5(1%2B3%2B3%5E2+...+3%5E%28n-1%29). Since
1%2B3%2B3%5E2+...+3%5E%28n-1%29=%283%5En+-+1%29%2F2,
then
f%5En%28x%29=%283%5En%29x-%285%2F2%29%283%5En+-+1%29, or
f%5En%28x%29=%283%5En%29%28x-5%2F2%29%2B5%2F2%29,